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Nested Allpass Filter Design

Any delay-element or delay-line inside a stable allpass-filter can be replaced by any stable allpass-filter to obtain a new stable allpass filter:

$\displaystyle z^{-1}\leftarrow H_a(z)\,z^{-1}
$

(The pure delay on the right-hand-side guarantees no delay-free loops are introduced, so that the original structure can be used)

Proof:

  1. Allpass Property: Note that the above substitution is a conformal map taking the unit circle of the $ z$ plane to itself. Therefore, unity gain for $ \vert z\vert=1$ is preserved under the mapping.
  2. Stability: Expand the transfer function in series form:

    $\displaystyle S\left([H_a(z)z^{-1}]^{-1}\right) \;=\;
s_0 + s_1 H_a(z)z^{-1}+ s_2 H_a^2(z)z^{-2}+ \cdots
$

    where $ s_n=$ original impulse response. In this form, it is clear that stability is preserved if $ H_a(z)$ is stable.


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``Computational Acoustic Modeling with Digital Delay'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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